Idea

To free the notion from membership-based set theory, we must replace sets of sets by families of sets, just as in passing from power sets to power objects we must replace sets of subsets by families of subsets.

Definition

A universe in a topos ℰ\mathcal{E} is a morphism el:E→Uel\colon E \to U satisfying the axioms to follow. We think of el:E→Uel\colon E \to U as an UU-indexed family of objects/sets (fibers of elel being those objects), and we define a morphism a:A→Ia\colon A \to I (regarded as an II-indexed family of objects) to be UU-small if there exists a morphism f:I→Uf\colon I \to U and a pullback square

The arrow ff is sometimes called the name of a:A→Ia\colon A \to I, since in the case when I=*I=*, the arrow f:*→Uf:* \to Upoints at the term in the universe UU representing the object AA. (See also this discussion and references there.)

Note that ff is not, in general, unique: a universe can contain many isomorphic sets. With this definition, the pullback of a UU-small morphism is automatically again UU-small. We say that an object XX is UU-small if X→1X\to 1 is UU-small.

Note that since 0→Ω0\to \Omega is a monomorphism, (1) and (4) imply that the initial object00 is UU-small. A predicative universe is a morphism el:E→Uel\colon E \to U where instead of (4) we assume merely that 00 is UU-small; this makes sense in any locally cartesian closed category. In a topos, the generic subobject 1→Ω1\to \Omega is a predicative universe, and of course a morphism is Ω\Omega-small if and only if it is a monomorphism.

If we assume only (1)–(3), then the identity morphism 10:0→01_0\colon 0 \to 0 of the initial object would be a universe, for which it itself is the only UU-small morphism. On the other hand, if ℰ\mathcal{E} has a natural numbers objectNN, we may additionally assume that NN is UU-small, to ensure that all universes contain “infinite” sets.

Note that any object isomorphic to a UU-small object is UU-small; thus in the language of Grothendieck universes this notion of smallness corresponds to essential smallness. Roughly, we may say that (1) corresponds to transitivity of a Grothendieck universe, (3) and (4) correspond to closure under power sets, and (2) corresponds to closure under indexed unions.

Example: universes in SETSET

We spell out in detail some implications of these axioms for the case that the topos in question is the Categeory of Sets according to ETCS, to be denoted SETSET.

Write ** for theterminal object in SETSET, the singleton set. Notice that for each ordinary element u∈Uu \in U, i.e. *→uU* \stackrel{u}{\to} U, there is the set EuE_u over uu, defined as the pullback

We think of EE as being the disjoint union over UU of the EuE_u. In the language of indexed categories, this is precisely the case: the object E∈SETE\in SET is the indexed coproduct of the UU-indexed family (E→U)∈SET/U(E\to U) \in SET/U.

By the definition of UU-smallness and the notation just introduced, an object SS in SETSET, regarded as a **-indexed family S→*S \to *, is UU-small precisely if it is isomorphic to one of the EuE_u.

If SS is a UU-small set by the above and if S0↪SS_0 \hookrightarrow S is a monomorphism so that S0S_0 is a subset of SS, it follows from 1) and 2) that the comoposite (S0↪S→*)=(S0→*)(S_0 \hookrightarrow S \to *) = (S_0 \to *) is UU-small, hence that S0S_0 is UU-small. So: a subset of a UU-small set is UU-small.

In particular, let ∅\emptyset be the initial object, which is a subset ∅↪Ω\emptyset \hookrightarrow \Omega of Ω=2\Omega = \mathbf{2}. So: the empty set is UU-small.

Let SS, TT and KK be objects of SETSET, regarded as **-indexed families f:S→*f\colon S \to *, T→*T \to * and K→*K \to *. Notice that (SET↓S)(f*K,f*T)≃(SET↓S)(K×S↓p2S,T×S↓p2S)(SET\downarrow S)(f^* K, f^* T) \simeq (SET\downarrow S)(\array{K \times S \\ \downarrow^{p_2} \\ S}, \array{T \times S \\ \downarrow^{p_2} \\ S}) is canonically isomorphic to SET(K×S,T)SET(K \times S, T). Since Πf\Pi_f is defined to be the right adjoint to f*:SET→SET↓Sf^*\colon SET \to SET \downarrow S it follows that Πff*T≃TS\Pi_f f^* T \simeq T^S is the function set of functions from SS to TT. By 3), if SS, TT are UU-small then so is the function set TST^S.

Since by 4) Ω=2\Omega = \mathbf{2} is UU-small and for every SS the function set 2S≃P(S)\mathbf{2}^S \simeq P(S) is the power set of SS, it follows that the power set of a UU-small set is UU-small.

Let II be a UU-small set, in that I→*I \to * is UU-small, and let S→IS \to I be UU-small, to be thought of as an II-indexed family of UU-small sets SiS_i, where SiS_i is the pullbackSi→S↓↓*→iI\array{
S_i &\to& S
\\
\downarrow && \downarrow
\\
* &\stackrel{i}{\to}& I
}, so that SS is the disjoint union of the SiS_i: S=sqcupi∈ISiS = sqcup_{i \in I} S_i. By axiom 2) the composite morphism (S→I→*)=(S→*)(S \to I \to *) = (S \to *) is UU-small, hence SS is a UU-small set, hence the II-indexed union of UU-small sets ⊔i∈ISi\sqcup_{i \in I} S_i is UU-small.

By standard constructions in set theory from these properties the following further closure properties of the universe UU follow.

For II a UU-small set and S→IS \to I an II-indexed family of UU-small sets SiS_i, the cartesian product ∏i∈ISi\prod_{i \in I} S_i is UU-small, as it is a subset of P(I×S)P(I \times S).

Axioms of universes

Just as ZFC and other material set theories may be augmented with axioms guaranteeing the existence of Grothendieck universes, so may ETCS and other structural set theories be augmented with axioms guaranteeing the existence of universes in the above sense. For example, the counterpart of Grothendieck’s axiom

For every set ss there exists a universe UU containing ss, i.e. s∈Us\in U

would be

For every morphism a:A→Ia\colon A \to I in ℰ\mathcal{E}, there exists a universe el:E→Uel\colon E \to U such that aa is UU-small.

Consequences

One can show, from the above axioms, that the UU-small morphisms are closed under finite coproducts and under quotient objects. See the reference below.

In terms of indexed categories

Recall that an ℰ\mathcal{E}-indexed category is a pseudofunctorℰop→Cat\mathcal{E}^{op}\to \Cat. The fundamental ℰ\mathcal{E}-indexed category is the self-indexing𝔼\mathbb{E} of ℰ\mathcal{E}, which takes I∈ℰI\in \mathcal{E} to the slice category𝔼I=ℰ/I\mathbb{E}^I = \mathcal{E}/I and x:I→Jx\colon I \to J to the base change functor x*x^*.

An internal full subcategory of ℰ\mathcal{E} is a full sub-indexed category 𝔽\mathbb{F} of 𝔼\mathbb{E} (that is, a collection of full subcategories 𝔽I⊂𝔼I\mathbb{F}^I\subset \mathbb{E}^I closed under reindexing) such that there exists a generic 𝔽\mathbb{F}-morphism, i.e. a morphism el:E→Uel\colon E \to U in 𝔽U\mathbb{F}^U such that for any a:A→Ia\colon A \to I in 𝔽I\mathbb{F}^I, we have a≅f*(el)a \cong f^*(el) for some f:I→Uf\colon I \to U. In this case (since ℰ\mathcal{E} is locally cartesian closed) there exists an internal categoryU1⇉UU_1 \;\rightrightarrows\; U in ℰ\mathcal{E} such that 𝔽\mathbb{F} is equivalent, as an indexed category, to the indexed category represented by U1⇉UU_1 \;\rightrightarrows\; U.

An internal full subcategory is an internal full subtopos if each 𝔽\mathbb{F} is a logical subtopos of 𝔼\mathbb{E} (closed under finite limits, exponentials, and containing the subobject classifier). A universe in ℰ\mathcal{E}, as defined above, can then be identified with an internal full subtopos satisfying the additional axiom that UU-small morphisms are closed under composition.

In the internal logic

In a topos with a universe, we can talk about small objects in the internal logic by instead talking about elements of UU. We can then rephrase the axioms of a universe in the internal logic to look more like the usual axioms for a Grothendieck universe, with the morphism el:E→Uel\colon E \to U interpreted as a “family of objects” (Su)u:U(S_u)_{u\colon U}:

for all uu in UU, if XX is a subset of SuS_u (in the sense that there exists an injectionX↪SuX \embedsin S_u), then there is a vv in UU such that X≅SvX \cong S_v;

for all uu in UU, there is a vv in UU such that the power setP(Su)≅SvP(S_u) \cong S_v;